On the Cyclicity of the Unramified Iwasawa Modules of the Maximal Multiple ℤ p -Extensions Over Imaginary Quadratic Fields

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For an odd prime number p, we study the number of generators of the unramified Iwasawa modules of the maximal multiple ℤ p -extensions over the Iwasawa algebra. In our previous paper, under several assumptions for an imaginary quadratic field, we obtained a necessary and sufficient condition for the cyclicity of the Iwasawa module over the Iwasawa algebra. The present work provides computational methods and numerical examples of Iwasawa modules that are cyclic as modules over the Iwasawa algebra. We remark that our methods do not require the assumption that Greenberg’s generalized conjecture holds.